NUCLEAR BINDING ENERGY, FISSION, FUSION

We saw that electrons are bound to nuclei in atoms and the mutual energy of interaction is negative. Binding energies are negative because it takes positive external work to pull atoms apart, i.e, to disassemble them into individual, separated electrons and nuclei. The single electron in the ground state of a hydrogen atom has energy -13.6 electron volts (eV).20 This means that 13.6eV * 1 ^[1]

of energy is required to remove the electron from being bound to the single proton nucleus.

Binding energies are about 10⁵ times larger for nuclei than for atoms. Protons attract other protons and neutrons, and neutrons attract one another, all by the strong nuclear force. Typically the binding energy per nucleon is 1 — 10 MeV. Nuclei are classified in terms of their number of protons z (atomic number) and total number of nucleons A.^[2] The number of neutrons is (A — z).

In the nucleus of He⁴, z = A — z = 2. The negative binding energy of helium-4 means that its mass differs from the masses of two protons and two neutrons outside the nucleus by Am < 0. The Einstein mass-energy relationship enables a calculation of the energy equivalent of any mass m, namely, with energy = me², where c = 2.9979246 x 10⁸ ms"¹ is the speed of light in a vacuum.

• - For an isolated proton, the mass is m_p = 1.6726219 x 10^_27kg and the corresponding energy is m_pc² = 938.27208 MeV.
• - The mass of an isolated neutron is m_n = 1.674927410 ²⁷ kg and the corresponding energy is m_nc² = 939.56541 MeV.
• - Adding the masses of two isolated neutrons and two isolated protons and calculating its energy equivalent gives total energy = 3755.675 MeV.
• - The mass of a ⁴He nucleus is 6.644657 x 10^-27 kg and the corresponding energy is 3737.379 MeV.
• - The difference is 3727.379 — 3755.675 = 28.3 MeV = BE the binding energy of ⁴He.

The nuclear force has interesting characteristics. (1) It is a very short range force, acting only over distances of about 10 ¹⁵ m For larger distances it is effectively zero. (2) The nuclear force saturates; i.e., as more nucleons are packed into a nucleus, the electrostatic repulsive force between protons grows but the binding energy per nucleon diminishes. Thus, high A nuclei are unstable. An informative graph of binding energy, |7?|/A, per nucleon is in Fig. 4.33. The lightest elements have the smallest |7?|, and |7?| = 0 (by definition) for hydrogen which has 0 = A = 1. As A increases, with a few exceptions, |JB|/A increases until nickel-62 (⁶²Ni), which is near the labeled ^5GFe, is reached. For A > 62, the envelope of the curve B/ A vs. A decreases from its maximum value of nearly 8.8 MeV per nucleon.

Key Point 4.35 Notably, for low atomic mass nuclei, fusion leads to higher atomic mass and larger binding energy magnitude. For high atomic mass nuclei, fission leads to smaller atomic masses and again, a larger binding energy magnitude. Total nuclear energy decreases in both fission and fusion.

Figure 4.33: The absolute value of the negative energy of binding per nucleon, |В|/А vs. A. Graphing the algebraic counterpart BE/A would turn the graph upside down. (Reprinted with permission by Fast fission.)

Entropy is produced in both fission and fusion, which are irreversible processes. The tendency to associate entropy with an increase with disorder associated with production of more particles in fission is misleading, because nuclear fusion decreases the number of particles and still generates entropy.

JARZYNSKI FREE ENERGY EQUALITY

NOTE: This section is more advanced mathematically than most of this monograph. To close this chapter, I return to the topic of free energy, discussed in Sec. 3.5. In 1997, Christopher Jarzynski proved a mathematical identity that has been influential in the field of non-equilibrium statistical mechanics.^[3] His result has become known as the Jarzynski equality, which relates a system’s Helmholtz free energy change between two given states to an average {e^-" /^T), where W is the work done on the system to effect the change.

with r = kT (see Eq. (3.14)). No single specific work path, e.g., fast or slow, is implied, but rather, all possible paths are contained in the averaging process, as explained here.

1. The angled brackets () connote an average over all possible experiments that can connect the two equilibrium states.

Figure 4.34: Expansion of an ideal gas, which can be fast, slow, and everything in between. All such processes are captured in the Jarzynski equality for an expansion from h, —> hj.

2. The paths are characterised by a parameter A(f), which at time t = 0 is A(0), and at the final time t = tf > 0 is A(iy). In the example in Fig. 4.34, A(0) = hi, the height of the gas at time zero and A(f/) = /г/, the final height at time t/. We envisage carrying out the same experiment numerous times, each time following a different path A(t). This results in a set of work values, {W. И'2,...}. The continuous probability distribution V{W) reflects the distribution of the large set of discrete work values, i.e., V{W)dW represents the fraction of experiments for which the measured work was in the interval (IT, W + dW).^[4]

3. Jarzynski used the probability distribution V(W) to define the left side of Eq. 4.76,

where the integral includes all possible work values.

The function / = e^_vl !^T is a convex function, for which all chords lie above (or on) /, all tangents lie below (or on) /, and the second derivative is nonnegative, as in Fig. 4.35. The average of / must he above / itself, i.e., in the shaded area, while f((W)/r) obviously lies on /. Jensen’s inequality guarantees that^[5]

Figure 4.35: The average (e ^W/^T) is in the shaded area, above e I^й )/^T, in accord with Jensen’s inequality, Eq. (4.78).

That this is reasonable is suggested by examples of the left and right sides of Jensen’s inequality. By Jarzynski’s equality, the left side of Eq. (4.78) equals е^-ЛЛ/т, and (4.78) becomes e~^AA/^T > e~(^w^^T, implying (W) > ДA. This and Eq. (4.76) generalise and extend Eq. 3.41, and are particularly useful for small systems, where thermal fluctuations make individual work values for a single experiment unreliable. To get a better sense of what this all means, consider two examples that are inspired by the work of Jarzynski.^[6] Macroscopic example. First we revisit the rubber band example introduced in Sec. 4.6. Here, let the rubber band be fixed to a left wall with a blob of glue, and subjected to an external force at its right end, as in Fig. 4.36(a). Initially, the rubber band is not stretched, but is taut, with length x_t. During the stretching process, the external force does work W on the rubber band, increasing its length from xi to xj (not shown). The process is carried out by choosing a function A(f) = x(t) and a positive time t/, with 0 < t < tf. These determine the path for given initial and final values ж(0) = A and x(t/) = В that governs the speed of the stretching process.

• 1. If the rubber band is stretched very slowly, i.e., with a slowly varying X(t) and large tf. the process is approximately isothermal because there is sufficient time for energy exchanges with the environment. It is also nearly reversible, with approximately the same resulting free energy difference ДA = Aj — A, for each repeated experiment. In this case, (IF) ss IF_rev = ДД, where the average includes only very slow expansions. Minimal work IF_rev is done and minimal energy is delivered to the surroundings.
• 2. For a relatively fast stretching process, more work is done, i.e., W > IF_rev, and there is insufficient time for energy transfer to the environment during the process, and the rubber band’s temperature increases. There is a larger energy transfer to the surroundings after the fast stretch than for the slow stretch.

Figure 4.36: (a) Stretching of a macroscopic rubber band, (b) Stretching of a long-chain molecule. The environment has fixed temperature T.

3. The Jarzynski equality, (e~^w/^T) = e~^AA/^T, entails an average over all possible stretches that connect states A and B. It turns out that for some problems, it is possible to evaluate the left side of the Jarzynski equality and thus obtain a value for the process-independent free energy change ДЛ = Ab — A a- We shall see with the next example, this averaging process is essential for sufficiently small systems, where thermal fluctuations are dominant .

Microscopic example. Figure 4.36(b) shows a similar example, with the rubber band replaced by a polymeric molecule that is visibly coiled, and can be stretched in principle, by attaching its ends to tiny beads. The left bead is held at the left end by a micropipette, and the right bead is in a laser trap that enables it to be moved left or right. Thermal fluctuations occur at both ends, making the initial length a’, uncertain.

For this microscopic system, use of the second law becomes questionable. Some quantities hold in an average sense, albeit with relatively large fluctuations about the average. As Jarzynski has described it, the system is immersed in aqueous solution at room temperature and pressure. The distance i = xj — Xi is varied by moving the centre of the optical trap. Initially, the system is in equilibrium with the environment. Work W is done on the molecule as the optical trap is moved rightward, following a prescribed path Л(t). After completion of the stretch, the system is allowed to equilibrate with the environment, with the pipette-to-trap distance held fixed at l.

If the experiment is repeated, thermal fluctuations of the long-chain molecule and of the molecules of the aqueous solution cause variations in the work required to stretch the molecule. Repetition gives a set of work values, {Wi, W2 ■ ■. }. The distribution of work values might look like Fig. 4.37(a). Typically, W > Д.4. but with a small probability that W < ДА The latter behaviour violates the second law of thermodynamics, occurs only rarely, and cannot be repeated at will. It is an example of Boltzmann’s observation that the entropy of an isolated system (a target system plus environment ) increases with a probability near, but less than, one.

Because of the substantial fluctuations in this microscopic system, the equation W > ДА fails to have meaning and must be replaced by

Figure 4.37: (a) Work distribution for: (a) a forward process, e.g., stretching a long-chain molecule, (b) Asymmetric forward and reversed processes, e.g., relaxation of a stretched molecule. The processes follow the protocol A(t) and A(—t) (time-reversed path), respectively.

This ends the discussion of the microscopic example. For the latter two examples, the Jarzynski equality is a strong statement that provides a strict equality (rather than an inequality) between average work behaviour and Helmholtz free energy change.

In addition to the Jarzynski equality, there are two other equalities that deserve mention. One is a work fluctuation theorem, proven by Gavin Crooks. It compares the probability distribution, Vp(W), that external work W is done on a system for the path г —> / using the protocol A(t), with the probability distribution Pr(W) that the conjugate process / —> i entails external work —W (on the system). The conjugate process follows the time-reversed protocol A(—t). The qualitative behaviour of the work distributions for both paths is shown in Fig. 4.37(b).

Key Point 4.36 The following points are notable.

• 1. Equation (3.24) is W - A A > 0, which implies TV (IF) > Pr(W). However because of the statistical nature of the second law of thermodynamics, that is not always true, but holds almost all the time. This is clear in Fig. 4-37(b). Such violations of the second law are rare, unrepeatable, and are fluctuations rather than average behaviour.
• 2. The asymmetry between the forward and reversed processes is evident in Fig. 4-37(h). It reflects the second law of thermodynamics, which is evident from the fact that. (Wp) + (Wr) > 0. If the latter inequality were reversed that would imply that the rubber band did positive work in a cyclic process, using energy from a single reservoir, which violates the Kelvin-Planck statement of the second law.

• 3. Figure 4-37(b) shows that when the work W = AA, the right side of (4.80) is one, and the forward and reverse distributions are equal.
• 4. It might seem that the inequality (№') > AA is a statement of the second law of thermodynamics. However, we have assumed processes that connect equilibrium states i and f. This implicitly assumes the assumption that the second law of thermodynamics holds.

A related expression for the probabilities of achieving total entropy change of the system plus environment, AStot. for forward and reverse processes was derived by Crooks,²⁰

When AStot > 0, P_F(AS_lol/k) > P/?(-A.9_tol/A"), and if AS_tot < 0, the reverse is true. A connection between (4.81) and (4.76) can be made using the fact that for process i —> /, the system’s internal energy change is AE = Q + W, the entropy change is AS, and the environment’s entropy change is —Q/T. The system’s change in the Helmholtz free energy is AA = AE — TAS = Q + W — TAS. This implies that the total entropy production is

Let oj = AStot/k temporarily, which leads to VF(co)e~^wI^T = e~^AA/^TVfi(—u!). Integrating both sides of the latter over (—oc < oj < oo), we have

Note that ш = W — A A/т. Each of the experiments in the ensemble of measurements has the same AA because the initial and final equilibrium states are the same. We can replace the variable ш by W/т with the same limits of integration. Thus, the left side of (4.83) is the average (e^-Vl/^T). The right side reduces to c~^AA/ so Eq. (4.81) for entropy fluctuations reduces to the Jarzynski equality (4.76) for work fluctuations.

Key Point 4.37 The mathematical arguments above illustrate that by considering ensembles of measurements, we can develop a framework for dealing with processes between equilibrium states that entail non-equilibrium paths. This formalism is suitable for small (non-macroscopic, and even microscopic) systems and irreversible processes. The energy-based Jarzynski equality (4.76) can be derived from, the entropy-based Crooks equality (4.81), revealing once again an intimate relationship between energy and entropy. ^[7]

The fluctuation equat ions of Jarzynski and Crooks have caused a considerable stir in the scientific community. Numerous articles have been published covering applications of these equations. This is because, they represent a new way to treat and understand non-equilibrium phenomena, which extends the applicability of traditional statistical mechanics.

• [1] 20leV= 1.6 x 10-19J, the energy gained by an electron that is accelerated through a volt electric potential difference.
• [2] This should be distinguished by context from the symbol A for the Helmholtz freeenergy.
• [3] C. Jarzynski, “Nonequilibrium Equality for Free Energy Differences,” Phys. Rev.Lett. 78. 2690-2693 (1997); C. Jarzynski, “Equilibrium free-energy differences fromnonequilibrium measurements: A master-equation approach,” Phys. Rev. E 556, 5018-5035(1997).
• [4] Theoretically, for a classical system, V(W) can be found in principle from thedistribution of trajectories in the бЛГ-dimensional phase space (3 momentum and 2 positioncomponents for each of N molecules), beginning with an ensemble of systems in a givenequilibrium macroscopic state. Each classical microstate will have its own initial state andwill evolve differently, in accord with Hamiltonian’s equations of motion.
• [5] Jensen’s inequality states that if X is a random variable and f(X) is a convex functionof X, then denoting by () an average over X, then {/(AT)) > f((X)).
• [6] C. Jarzynski, “Nonequilibrium work relations: foundations and applications,” TheEuropean Physical Journal B64. 331-340 (2008).
• [7] G. E. Crooks, “Entropy production fluctuation theorem and the nonequilibrium workrelation for free energy differences,” Phys. Rev. E 60, 2721-2726 (1999). Crooks useddifferent notation. The ratio Sioi/k is used here because it is dimensionless.